Question

You are given that sin(A)=817, with A in Quadrant II, and cos(B)=−513, with B in Quadrant II. Find cos(A−B). Give your answer as a fraction.

Answer 1

To find cos(A-B), apply the cosine difference identity using sin(A) = 8/17 and cos(B) = -5/13, calculating the required cos(A) and sin(B) using the Pythagorean identity, considering that A and B are in Quadrant II.

Step-by-step explanation:

The question asks to find the cosine of the difference between two angles, A and B, given the sine of A and the cosine of B, with both angles in Quadrant II. To find cos(A-B), we can use the cosine difference identity, which states: cos(A − B) = cos A cos B + sin A sin B.

Since we are given sin(A) = 8/17, we can find cos(A) using the Pythagorean identity: cos² A = 1 - sin² A. Given that A is in Quadrant II, where cosine is negative, we can find cos(A) = - √(1 - (8/17)²).

Similarly, we are given cos(B) = -5/13 and since B is in Quadrant II, sin(B) is positive, so we can find sin(B) using the identity: sin² B = 1 - cos² B, sin(B) = √(1 - (-5/13)²).

Now, plug in the values into the identity: cos(A-B) = (- √(1 - (8/17)²))(-5/13) + (8/17)(√(1 - (-5/13)²)) to get the answer in fraction form.